Eigenvalues and Eigenvectors of the Matrix

The determination of eigenvalues and eigenvectors of the matrix A is based on a routine by Grad and Brebner (1968). The matrix is first scaled by a sequence of similarity transformations and then normalized to have the Euclidian norm equal to one. The matrix is reduced to an upper Hessenberg form by Householder s method. Then the QR double-step iterative process is performed on the Hessenberg matrix to compute the eigenvalues. The eigenvectors are obtained by inverse iteration. 

The eigenvalues and eigenvectors of the matrix B can be used to directly compute the new ellipticity R and orientation p in the stretched state. Carrying out the matrix multiplication B = PAP, the components of B are found to be given by ... 

Below a Matlab script for the calculation of a quadrature approximation of order N from a known set of moments iti using the Wheeler algorithm is reported. The script computes the intermediate coefficients sigma and the jacobi matrix, and, as for the PD algorithm, determines the nodes and weights of the quadrature approximation from the eigenvalues and eigenvectors of the matrix. 

EXAMPLE 10.3 Use Mathematica to find the eigenvalues and eigenvectors of the matrix in the previous example. 

We next solve the secular equation F —1 = 0 to obtain the eigenvalues and eigenvectors of the matrix F. This step is usually performed using matrix diagonalisation, as outlined in Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of these eigenvalues will be zero as they correspond to translational and rotational motion of the entire system. The frequency of each normal mode is then calculated from the eigenvalues using the relationship ... 

Scriven [334] showed that the stability of spatially discrete homogeneous reaction-diffusion systems can be analyzed in terms of the structural modes of the network, i.e., the eigenvectors of the Laplacian matrix L. We have extended that approach [305], and the eigenvalues and eigenvectors of the matrix... 

In the general case, the set of Equation 51.6 is solved by differential methods. If a step of reactant adsorption is irreversible, then the equations that describe the changes in isotope transfer in the gas phase are solved analytically. Further solution of the primal problem is based on the numerical search of eigenvalues and eigenvectors of the matrix at the right side of the equations describing changes of isotope fraction in the intermediate substances. In this case, the procedure of numerical analysis is performed much faster. 

If you are familiar with matrix algebra, note that solving (9.83) and (9.81) amounts to finding the eigenvalues and eigenvectors of the matrix whose elements are... 

Since (A-3) depends only on eigenvalues and eigenvectors of the matrix A one sees that it obtains irrespective of the initial distribution. In the case where the only non-zero matrix elements are the nearest-neighbor transition probabilities, (A 3) reduces to the following equation ... 

The solution of the set of linear ordinary differential equations is very cumbersome to evaluate in the form of Eq. (5.40), because it requires the evaluation of the inOnite series of the exponential term e. However, this solution can be modified by further algebraic manipulation to express it in terms of the eigenvalues and eigenvectors of the matrix A. 

---